Consider the functor \[h_2 \colon \mathrm{Alg}_{\mathbb E_2}(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]) \rightarrow\mathrm{Alg}_{\mathbb E_2}(\mathrm{Cat}_{({2}, {2})})\] induced by the lax symmetric monoidal composite \[\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}] \xrightarrow{\mathrm{Cat}[\mathrm{forget}]} \mathrm{Cat}[\mathrm{Cat}_{\infty}] = \mathrm{Cat}_{(\infty, {2})} \xrightarrow{h_2} \mathrm{Cat}_{({2}, {2})}\] which first forgets along \(\mathrm{st}^{B\mathbb{Z}}_{k}\rightarrow\mathrm{Cat}_{\infty}\)38 the homwise structure and then takes the homotopy \(2\)-category (definition 5.4.11). Applying this to \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Alg}_{\mathbb E_2}(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}])\) results in the braided monoidal \((2,2)\)-category \(\mathcal H\coloneqq h_2{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Alg}_{\mathbb E_2}(\mathrm{Cat}_{({2}, {2})})\) described in subsection 1.1.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2